Section 8.4 Nack/Jones 1

Section 8.4Polyhedrons & Spheres

Section 8.4 Nack/Jones 2

Polyhedron

• Plural: polyhedrons or polyhedra• A solid bounded by plane regions.• The faces of the polyhedrons are polygons• The edges are the line segments common to these

polygons • Vertices are the endpoints of the edges• Convex: Each face determines a plane for which all

remaining faces lie on the same side of the plane. p.420.

• Concave: Two vertices and the line segment containing them lies in the exterior of the polyhedron.

Section 8.4 Nack/Jones 3

Euler’s Equation

• Theorem 8.4.1: The number of vertices V, the number of edges, E, and the number of faces F of a polyhedron are related by the equation.

V + F = E + 2

Where V = # of vertices F = # of faces E = # of edges

Example 1 p. 420

Section 8.4 Nack/Jones 4

Regular Polyhedron• A regular polyhedron is a convex polyhedron

whose faces are congruent regular polygons arranged in such a way that adjacent faces form congruent dihedral angles (the angle formed when two edges intersect).

Section 8.4 Nack/Jones 5

Spheres

• Three Characteristics

1. A sphere is the set of all points

at a fixed distance r from a given

point O. Point O is known as the

center of the sphere.

2. A sphere is the surface determined when a circle

(or semicircle) is rotated about any of its diameters.

3. A sphere is the surface that represents the theoretical

limit of an “inscribed” regular polyhedron whose number

of faces increase without limit.

Section 8.4 Nack/Jones 6

Surface Area and Volume of a Sphere

• Theorem 8.4.2: The surface area S

of a sphere whose radius has length r

is given by S = 4r²

• Theorem 8.4.3: The volume V of a

sphere with radius of length r is given by V =4/3 r3

Example 4 – 6 p. 424• Solids of Revolution:

– Revolving a semi circle = sphere– Revolving circle around line = torus p. 425 - 426